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Those Fascinating Numbers
Page58(78 of 451)

58 Jean-Marie De Koninck x N(x) 10 10 100 33 1000 213 x N(x) 104 1 538 105 11 872 106 95 428 x N(x) 107 806 095 108 6 954 793 109 61 574 510 • the smallest number n such that λ0(n) = λ0(n + 1) = . . . = λ0(n + 7) = 1, where λ0 stands for the Liouville function; moreover, in this case, we also have λ0(n + 8) = 1 (see the number 1 934 for the list of the smallest numbers at which the λ0 function takes the value 1 at consecutive arguments). 215 • the second solution of σ2(n) = σ2(n + 2) (see the number 1 079). 216 • the only cube which can be written as the sum of the cubes of three consecutive numbers: 216 = 63 = 33 + 43 + 53; • the tenth 3-powerful number (counting 1 as a 3-powerful number); if nk stands for the kth 3-powerful number, then n10 = 216, n100 = 52 488, n1 000 = 25 153 757, n10 000 = 16 720 797 973, n100 000 = 13 346 039 198 336 and n1 000 000 = 11 721 060 349 748 875. 217 • the smallest number 1 for which the sum of its divisors is a fourth power: σ(217) = 44; the sequence of numbers satisfying this property begins as follows: 217, 510, 642, 710, 742, 782, 795, 862, 935, . . . ; it is also the smallest number n such that σ(n) is an eighth power: σ(217) = 28. 220 • the number which, when paired with the number 284, forms the smallest ami- cable pair: two numbers are called amicable if the sum of the proper divisors of one of them equals the other: here σ(220) − 220 = 284 and σ(284) − 284 = 220. 221 (= 102 + 112 = 52 + 142) • the number used by Euler to show how to find its factors given that two distinct representations of it as the sum of two squares70 are known. 70Here is Euler’s method. Let n be an odd number which can be written as the sum of two squares in two distinct ways: n = a2 + b2 = c2 + d2,